Chapter 2. Hypothesis testing in one population


 Debra Henry
 2 years ago
 Views:
Transcription
1 Chapter 2. Hypothesis testing in one population Contents Introduction, the null and alternative hypotheses Hypothesis testing process Type I and Type II errors, power Test statistic, level of significance and rejection/acceptance regions in upper, lower and twotail tests Test of hypothesis: procedure pvalue Twotail tests and confidence intervals Examples with various parameters Power and sample size calculations
2 Chapter 2. Hypothesis testing in one population Learning goals At the end of this chapter you should be able to: Perform a test of hypothesis in a onepopulation setting Formulate the null and alternative hypotheses Understand Type I and Type II errors, define the significance level, define the power Choose a suitable test statistic and identify the corresponding rejection region in upper, lower and twotail tests Use the pvalue to perform a test Know the connection between a twotail test and a confidence interval Calculate the power of a test and identify a sample size needed to achieve a desired power
3 Chapter 2. Hypothesis testing in one population References Newbold, P. Statistics for Business and Economics Chapter 9 ( ) Ross, S. Introduction to Statistics Chapter 9
4 Test of hypothesis: introduction A test of hypothesis is a procedure that: is based on a data sample and allows us to make a decision about a validity of some conjecture or hypothesis about the population X, typically the value of a population parameter θ (θ can be any of the parameters we covered so far: µ, p, σ 2, etc) This hypothesis, called a null hypothesis (H 0 ): Can be thought of as a hypothesis being supported (before the test is carried out) Will be believed unless sufficient contrary sample evidence is produced When sample information is collected, this hypothesis is put in jeopardy, or tested
5 The null hypothesis: examples 1. A manufacturer who produces boxes of cereal claims that, on average, their contents weigh at least 20 ounces. To check this claim, the contents of a random sample of boxes are weighed and inference is made. Population: X = weight of a box of cereal (in oz) µ 0 z} { Null hypothesis, H 0 : µ 20 'SRS Does sample data produce evidence against H 0? 2. A company receiving a large shipment of parts accepts their delivery only if no more than 50% of the parts are defective. The decision is based on a check of a random sample of these parts. Population: X = 1 if a part is defective and 0 otherwise X Bernoulli(p), p = proportion of defective parts in the entire shipment p 0 z} { Null hypothesis, H 0 : p 0.5'SRS Does sample data produce evidence against H 0?
6 Null hypothesis, H 0 States the assumption to be tested We begin with the assumption that the null hypothesis is true (similar to the notion of innocent until proven guilty) Refers to the status quo Always contains a =, or sign (closed set) May or may not be rejected Simple hypothesis (specifies a single value): H 0 : µ = µ 0 z} { 5, H 0 : p = p 0 z} { 0.6, H 0 : σ 2 = Parameter space under this null: Θ 0 = {θ 0} Composite hypothesis (specifies a range of values): H 0 : µ σ 2 0 z} { 9 In general: H 0 : θ = θ 0 µ 0 p 0 z} { z} { 5, H 0 : p 0.6 In general: H 0 : θ θ 0 or H 0 : θ θ 0 Parameter space under this null: Θ 0 = (, θ 0] or Θ 0 = [θ 0, )
7 Alternative hypothesis, H 1 If the null hypothesis is not true, then some alternative must be true, and in carrying out a hypothesis test, the investigator formulates an alternative hypothesis against which the null hypothesis is tested. The alternative hypothesis H 1: Is the opposite of the null hypothesis Challenges the status quo Never contains =, or sign May or may not be supported Is generally the hypothesis that the researcher is trying to support Onesided hypothesis: (uppertail) H 1 : µ > 5 (lowertail) H 0 : p < 0.6 In general: H 1 : θ > θ 0 or H 1 : θ < θ 0 Parameter space under this alternative: Θ 1 = (θ 0, ) or Θ 1 = (, θ 0) Twosided hypothesis (twotail): H 1 : σ 2 9 In general: H 1 : θ θ 0 Parameter space under this alternative: Θ 1 = (, θ 0) (θ 0, )
8 The alternative hypothesis: examples 1. A manufacturer who produces boxes of cereal claims that, on average, their contents weigh at least 20 ounces. To check this claim, the contents of a random sample of boxes are weighed and inference is made. Population: X = weight of a box of cereal (in oz) Null hypothesis, H 0 : µ 20 versus Alternative hypothesis, H 1 : µ < 20'SRS Does sample data produce evidence against H 0 in favour of H 1? 2. A company receiving a large shipment of parts accepts their delivery only if no more than 50% of the parts are defective. The decision is based on a check of a random sample of these parts. Population: X = 1 if a part is defective and 0 otherwise X Bernoulli(p), p = proportion of defective parts in the entire shipment Null hypothesis, H 0 : p 0.5 versus Alternative hypothesis, H 1 : p > 0.5'SRS Does sample data produce evidence against H 0 in favour of H 1?
9 Hypothesis testing process xyyxxxxyy Population: X = height of a UC3M student (in m) Claim: On average, students are shorter than 1.6 Hypotheses: H 0 : µ 1.6 versus H 1 : µ > 1.6 'SRS yyxx Sample: Suppose the sample mean height is 1.65 m, x = 1.65 Is it likely to observe a sample mean x = 1.65 if the population mean is µ 1.6? If not likely, reject the null hypothesis in favour of the alternative.
10 Hypothesis testing process Having specified the null and alternative hypotheses and collected the sample information, a decision concerning the null hypothesis (reject or fail to reject H 0 ) must be made. The decision rule is based on the value of a distance between the sample data we have collected and those values that would have a nigh probabiilty under the null hypothesis. This distance is calculated as the value of a socalled test statistic (closely related to the pivotal quantities we talked about in Chapter 1). We will discuss specific cases later on. However, whatever decision is made, there is some chance of reaching an erroneous conclusion about the population parameter, because all that we have available is a sample and thus we cannot know for sure if the null hypothesis is true or not. There are two possible states of nature and thus two errors can be committed: Type I and Type II errors.
11 Type I and Type II errors, power Type I Error: to reject a true null hypothesis. A Type I error is considered a serious type of error. The probability of a Type I Error is equal to α and is called the significance level. α = P(reject the null H 0 is true) Type II Error: to fail to reject a false null hypothesis. The probability of a Type II Error is β. β = P(fail to reject the null H 1 is true) power: is the probability of rejecting a null hypothesis (that is false). power = 1 β = P(reject the null H 1 is true) Actual situation Decision H 0 true H 0 false Do not No error Type II Error Reject H 0 (1 α) (β) Reject Type I error No Error H 0 (α) (1 β = power)
12 Type I and Type II errors, power Type I and Type II errors can not happen at the same time Type I error can only occur if H0 is true Type II error can only occur if H0 is false If the Type I error probability (α), then the Type II error probability β All else being equal: β when the difference between the hypothesized parameter value and its true value β when α β when σ β when n The power of the test increases as the sample size increases For θ Θ1 power(θ) = 1 β For θ Θ0 power(θ) α
13 Test statistic, level of significance and rejection region Test statistic, T Allows us to decide if the sample data is likely or unlikely to occur, assuming the null hypothesis is true. It is the pivotal quantity from Chapter 1 calculated under the null hypothesis. The decision in the test of hypothesis is based on the observed value of the test statistic, t. The idea is that, if the data provide an evidence against the null hypothesis, the observed test statistic should be extreme, that is, very unusual. It should be typical otherwise. In distinguishing between extreme and typical we use: the sampling distribution of the test statistic the significance level α to define socalled rejection (or critical) region and the acceptance region.
14 Test statistic, level of significance and rejection region Rejection region (RR) and acceptance region (AR) in size α tests: Uppertail test H 1 : θ > θ 0 α RR α = {t : t > T α} AR α = {t : t T α} AR CRITICAL VALUE RR Lowertail test H 1 : θ < θ 0 α RR α = {t : t < T 1 α} AR α = {t : t T 1 α} RR CRITICAL VALUE AR Twotail test H 1 : θ θ 0 RR α = {t : t < T 1 α/2 or t > T α/2 } AR α = {t : T 1 α/2 t T α/2 } α 2 α 2 RRCRITICAL AR CRITICALRR VALUE VALUE
15 Test statistics Let X n be a s.r.s. from a population X with mean µ and variance σ 2, α a significance level, z α the upper α quantile of N(0,1), µ 0 the population mean under H 0, etc. Parameter Assumptions Test statistic RRα in twotail test Mean Variance Normal data Known variance Nonnormal data Large sample Bernoulli data Large sample Normal data Unknown variance Normal data X µ 0 σ/ N(0, 1) n X µ 0 ˆσ/ ap. N(0, 1) n ˆp p 0 p p0 (1 p 0 )/n ap. N(0, 1) jz : X µ 0 s/ n t n 1 (n 1)s 2 σ 2 0 χ 2 n 1 >< χ 2 : 8 z 9 z } { >< x µ 0 z : σ/ < z 1 α/2 or x µ >= 0 n σ/ n > z α/2 >: >; j x µ z : 0 ˆσ/ n < z 1 α/2 or x µ ff 0 ˆσ/ n > z α/2 ff ˆp p p 0 p0 (1 p 0 )/n < z 1 α/2 or ˆp p p 0 p0 (1 p 0 )/n > z α/2 8 t 9 z } { >< x µ 0 t : s/ < t n 1;1 α/2 or x µ >= 0 n s/ n > t n 1;α/2 >: >; 8 9 χ 2 z } { (n 1)s 2 σ 2 0 < χ 2 (n 1)s2 or n 1;1 α/2 σ 0 2 > χ 2 n 1;α/2 >= St. dev. Normal data (n 1)s 2 σ 2 0 χ 2 n 1 >: ( χ 2 : (n 1)s 2 σ 2 0 >; ) < χ 2 (n 1)s2 or n 1;1 α/2 σ 0 2 > χ 2 n 1;α/2 Question: How would you define RR α in upper and lowertail tests?
16 Test of hypothesis: procedure 1. State the null and alternative hypotheses. 2. Calculate the observed value of the test statistic (see the formula sheet). 3. For a given significance level α define the rejection region (RR α ). Reject H0, the null hypothesis, if the test statistic is in RR α and fail to reject H 0 otherwise. 4. Write down the conclusions in a sentence.
17 Uppertail test for the mean, variance known: example Example: 9.1 (Newbold) When a process producing ball bearings is operating correctly, the weights of the ball bearings have a normal distribution with mean 5 ounces and standard deviation 0.1 ounces. The process has been adjusted and the plant manager suspects that this has raised the mean weight of the ball bearings, while leaving the standard deviation unchanged. A random sample of sixteen bearings is selected and their mean weight is found to be ounces. Is the manager right? Carry out a suitable test at a 5% level of significance. Population: X = weight of a ball bearing (in oz) X N(µ, σ 2 = ) Test statistic: Z = X µ 0 σ/ N(0, 1) n Observed test statistic: 'SRS: n = 16 Sample: x = Objective: test µ 0 z} { H 0 : µ = 5 against H 1 : µ > 5 (Uppertail test) σ = 0.1 µ 0 = 5 n = 16 x = z = x µ0 σ/ n = / 16 = 1.52
18 Uppertail test for the mean, variance known: example Example: 9.1 (cont.) Rejection (or critical) region: RR 0.05 = {z : z > z 0.05 } = {z : z > 1.645} z= 1.52 Since z = 1.52 / RR 0.05 we fail to reject H 0 at a 5% significance level. N(0,1) density AR z α = Conclusion: The sample data did not provide sufficient evidence to reject the claim that the average weight of the bearings is 5oz. RR
19 Definition of pvalue It is the probability of obtaining a test statistic at least as extreme ( or ) as the observed one (given H 0 is true) Also called the observed level of significance It is the smallest value of α for which H 0 can be rejected Can be used in step 3) of the testing procedure with the following rule: If pvalue < α, reject H0 If pvalue α, fail to reject H0 Roughly: small pvalue  evidence against H0 large pvalue  evidence in favour of H0
20 pvalue pvalue when t is the observed value of the test statistic T : Uppertail test H 1 : θ > θ 0 test stat p value =area pvalue = P(T t) Lowertail test H 1 : θ < θ 0 pvalue = P(T t) Twotail test H 1 : θ θ 0 pvalue = P(T t ) + P(T t ) p value =area test stat test stat p value =left+right areas test stat
21 pvalue: example Example: 9.1 (cont.) Population: X = weight of a ball bearing (in oz) X N(µ, σ 2 = ) 'SRS: n = 16 Sample: x = Objective: test µ 0 z} { H 0 : µ = 5 against H 1 : µ > 5 (Uppertail test) Test statistic: Z = X µ 0 σ/ N(0, 1) n Observed test statistic: z = 1.52 N(0,1) density pvalue = P(Z z) = P(Z 1.52) = where Z N(0, 1) Since it holds that pvalue = α = 0.05 we fail to reject H 0 (but would reject at any α greater than , e.g., α = 0.1). z= 1.52 p value =area
22 The pvalue and the probability of the null hypothesis 1 The pvalue: is not the probability of H0 nor the Type I error α; but it can be used as a test statistic to be compared with α (i.e. reject H 0 if pvalue < α). We are interested in answering: How probable is the null given the data? Remember that we defined the pvalue as the probability of the data (or values even more extreme) given the null. We cannot answer exactly. But under fairly general conditions and assuming that if we had no observations Pr(H 0) = Pr(H 1) = 1/2, then for pvalues, p, such that p < 0.36: ep ln(p) Pr(H 0 Observed Data) 1 ep ln(p). 1 Selke, Bayarri and Berger, The American Statistician, 2001
23 The pvalue and the probability of the null hypothesis This table helps to calibrate a desired pvalue as a function of the probability of the null hypothesis: pvalue Pr(H 0 Observed Data) For a pvalue equal to 0.05 the null has a probability of at least 29% of being true While if we want the probability of the null being true to be at most 5%, the pvalue should be no larger than
24 Confidence intervals and twotail tests: duality A twotail test of hypothesis at a significance level α can be carried out using a (twotail) 100(1 α)% confidence interval in the following way: 1. State the null and twosided alternative H 0 : θ = θ 0 against H 1 : θ θ 0 2. Find a 100(1 α)% confidence interval for θ 3. If θ 0 doesn t belong to this interval, reject the null. If θ 0 belongs to this interval, fail to reject the null. 4. Write down the conclusions in a sentence.
25 Twotail test for the mean, variance known: example Example: 9.2 (Newbold) A drill is used to make holes in sheet metal. When the drill is functioning properly, the diameters of these holes have a normal distribution with mean 2 in and a standard deviation of 0.06 in. To check that the drill is functioning properly, the diameters of a random sample of nine holes are measured. Their mean diameter was 1.95 in. Perform a twotailed test at a 5% significance level using a CIapproach. Population: 100(1 α)% = 95% confidence X = diameter of a hole (in inches) interval for µ: X N(µ, σ 2 = ) ( x 1.96 n σ ) 'SRS: n = 9 Sample: x = 1.95 Objective: test µ 0 {}}{ H 0 : µ = 2 against H 1 : µ 2 (Twotail test) CI 0.95 (µ) = = ( ) 9 = (1.9108, ) Since µ 0 = 2 / CI 0.95 (µ) we reject H 0 at a 5% significance level.
26 Twotail test for the proportion: example Example: 9.6 (Newbold) In a random sample of 199 audit partners in U.S. accounting firms, 104 partners indicated some measure of agreement with the statement: Cash flow from operations is a valid measure of profitability. Test at the 10% level against a twosided alternative the null hypothesis that onehalf of the members of this population would agree with the preceding statement. Population: X = 1 if a member agrees with the Test statistic: statement and 0 otherwise Z = ˆp p 0 X Bernoulli(p) approx. N(0, 1) p0(1 p 0)/n Observed test statistic: 'SRS: n = 199 large n Sample: ˆp = = Objective: test p 0 {}}{ H 0 : p = 0.5 against H 1 : p 0.5 (Twotail test) p 0 = 0.5 n = 199 ˆp = z = ˆp p 0 p0 (1 p 0 )/n = (1 0.5)/199 = 0.65
27 Twotail test for the proportion: example Example: 9.6 (cont.) Rejection (or critical) region: RR 0.10 = {z : z > z 0.05 } {z : z < z 0.05 } = {z : z > 1.645} {z : z < 1.645} z= 0.65 Since z = 0.65 / RR 0.10 we fail to reject H 0 at a 10% significance level. N(0,1) density RR z α 2 = 1.645AR zα = RR 2 Conclusion: The sample data does not contain sufficiently strong evidence against the hypothesis that onehalf of all audit partners agree that cash flow from operations is a valid measure of profitability.
28 Lowertail test for the mean, variance unknown: example Example: 9.4 (Newbold, modified) A retail chain knows that, on average, sales in its stores are 20% higher in December than in November. For a random sample of six stores the percentages of sales increases were found to be: 19.2, 18.4, 19.8, 20.2, 20.4, Assuming a normal population, test at a 10% significance level the null hypothesis (use a pvalue approach) that the true mean percentage sales increase is at least 20, against a onesided alternative. Population: X = stores increase in sales from Nov to Dec (in %s) X N(µ, σ 2 ) σ 2 unknown 'SRS: n = 6 small n Sample: x = = 19.5 s 2 = (19.5)2 6 1 = Objective: test µ 0 z} { H 0 : µ 20 against H 1 : µ < 20 (Lowertail test) Test statistic: T = X µ 0 s/ n tn 1 Observed test statistic: µ 0 = 20 n = 6 x = 1.95 s = = t = x µ0 s/ n = / 6 = 1.597
29 Lowertail test for the mean, variance unknown: example Example: 9.4 (cont.) pvalue = P(T 1.597) (0.05, 0.1) because t 5;0.05 t 5;0.10 { }} { { }} { < < Hence, given that pvalue < α = 0.1 we reject the null hypothesis at this level. p value =area t= t n 1 density Conclusion: The sample data gave enough evidence to reject the claim that the average increase in sales was at least 20%. pvalue interpretation: if the null hypothesis were true, the probability of obtaining such sample data would be at most 10%, which is quite unlikely, so we reject the null hypothesis.
30 Lowertail test for the mean, variance unknown: example Example: 9.4 (cont.) in Excel: Go to menu: Data, submenu: Data Analysis, choose function: twosample ttest with unequal variances. Column A (data), Column B (n repetitions of µ 0 = 20), in yellow (observed t stat, pvalue and t n 1;α ).
31 Uppertail test for the variance: example Example: 9.5 (Newbold) In order to meet the standards in consignments of a chemical product, it is important that the variance of their percentage impurity levels does not exceed 4. A random sample of twenty consignments had a sample quasivariance of 5.62 for impurity level percentages. a) Perform a suitable test of hypothesis (α = 0.1). b) Find the power of the test. What is the power at σ 2 1 = 7? c) What sample size would guarantee a power of 0.9 at σ 2 1 = 7? Population: X = impurity level of a consignment of a chemical (in %s) X N(µ, σ 2 ) Test statistic: χ 2 = (n 1)s2 σ0 2 Observed test statistic: χ 2 n 1 'SRS: n = 20 Sample: s 2 = 5.62 Objective: test H 0 : σ 2 σ0 2 z} { 4 against H 1 : σ 2 > 4 (Uppertail test) σ 2 0 = 4 n = 20 s 2 = 5.62 χ 2 = (n 1)s2 σ0 2 = (20 1) =
32 Uppertail test for the variance: example Example: 9.5 a) (cont.) pvalue = P(χ ) (0.1, 0.25) because χ 2 19;0.25 {}}{ 22.7 < < χ 2 19;0.1 {}}{ 27.2 Hence, given that pvalue exceeds α = 0.1, we cannot reject the null hypothesis at this level. χ 2 n 1 density χ 2 = p value =area Conclusion: The sample data did not provide enough evidence to reject the claim that the variance of the percentage impurity levels in consignments of this chemical is at most 4.
33 Uppertail test for the variance: power Example: 9.5 b) Recall that: power = P(reject H 0 H 1 is true) When do we reject H 0? j ff (n 1)s 2 RR 0.1 = > χ 2 σ0 2 n 1;0.1 power(σ 2 ) versus σ = >< z } { >= = (n 1)s 2 > χ 2 n 1;0.1 σ0 2 >: >; Hence the power is: power(σ1) 2 = P reject H 0 σ 2 = σ1 2 = P (n 1)s 2 > σ 2 = σ1 2 (n 1)s 2 = P > «σ1 2 σ1 2 = P χ 2 > ««108.8 = 1 F χ 2 σ 2 1 σ α (F χ 2 is the cdf of χ 2 n 1) Hence, power(7) = P `χ 2 > σ 0 2 = 4 power(σ 2 = 1 β(σ 2 ) Θ 0 Θ 1 σ =
34 Uppertail test for the variance: sample size calculations Example: 9.5 c) From our previous calculations, ( we know that ) potencia(σ1 2) = P (n 1)s 2 > χ 2 σ 2 0 n 1;0.1, σ 2 1 Our objective is to find the smallest n such that: σ 2 1 (n 1)s 2 σ 2 1 χ 2 n {}}{ (n 1)s 2 power(7) = P > χ 2 4 n 1; σ 2 1 The last equation implies that we are dealing with a χ 2 n 1 distribution, whose upper 0.9quantile satisfies χ 2 n 1; χ2 n 1;0.1. chisquare χ table 2 43;0.9 /χ2 43;0.1 = > n 1 = 43 Thus, if we collect 44 observations we should be able to detect the alternative value σ1 2 = 7 with at least 90% chance.
35 Another power example: lowertail test for the mean, normal population, known σ α H 0 : µ µ 0 versus H 1 : µ < µ 0 at α = 0.05 Say that µ 0 = 5, n = 16, σ = 0.1 We reject H 0 if x µ0 σ/ n < z α = that is when x 4.96, hence ( ) power(µ 1 ) = P Z < 4.96 µ1 0.1/ 16 power(µ) = 1 β(µ) µ 0 = Θ 0 Θ 1 µ n=16 n=9 n=4
36 Another power example: lowertail test for the mean, normal population, known σ 2 Note that the power = 1 P(Type II error) function has the following features (everything else being equal): The farther the true mean µ 1 from the hypothesized µ 0, the greater the power The smaller the α, the smaller the power, that is, reducing the probability of Type I error will increase the probability of Type II error The larger the population variance, the lower the power (we are less likely to detect small departures from µ 0, when there is greater variability in the population) The larger the sample size, the greater the power of the test (the more info from the population, the greater the chance of detecting any departures from the null hypothesis).
Chapter 8 Introduction to Hypothesis Testing
Chapter 8 Student Lecture Notes 81 Chapter 8 Introduction to Hypothesis Testing Fall 26 Fundamentals of Business Statistics 1 Chapter Goals After completing this chapter, you should be able to: Formulate
More informationHYPOTHESIS TESTING: POWER OF THE TEST
HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,
More informationChapter 9: Hypothesis Tests of a Single Population
Chapter 9: Hypothesis Tests of a Single Population Department of Mathematics Izmir University of Economics Week 12 20142015 Introduction In this chapter we will focus on Example developing hypothesis
More information7 Hypothesis testing  one sample tests
7 Hypothesis testing  one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X
More informationChapter 8. Hypothesis Testing
Chapter 8 Hypothesis Testing Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing
More informationHypothesis Testing  II
3σ 2σ +σ +2σ +3σ Hypothesis Testing  II Lecture 9 0909.400.01 / 0909.400.02 Dr. P. s Clinic Consultant Module in Probability & Statistics in Engineering Today in P&S 3σ 2σ +σ +2σ +3σ Review: Hypothesis
More information93.4 Likelihood ratio test. NeymanPearson lemma
93.4 Likelihood ratio test NeymanPearson lemma 91 Hypothesis Testing 91.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental
More informationChapter 9, Part A Hypothesis Tests. Learning objectives
Chapter 9, Part A Hypothesis Tests Slide 1 Learning objectives 1. Understand how to develop Null and Alternative Hypotheses 2. Understand Type I and Type II Errors 3. Able to do hypothesis test about population
More information3.4 Statistical inference for 2 populations based on two samples
3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted
More informationLecture 13 More on hypothesis testing
Lecture 13 More on hypothesis testing Thais Paiva STA 111  Summer 2013 Term II July 22, 2013 1 / 27 Thais Paiva STA 111  Summer 2013 Term II Lecture 13, 07/22/2013 Lecture Plan 1 Type I and type II error
More information[Chapter 10. Hypothesis Testing]
[Chapter 10. Hypothesis Testing] 10.1 Introduction 10.2 Elements of a Statistical Test 10.3 Common LargeSample Tests 10.4 Calculating Type II Error Probabilities and Finding the Sample Size for Z Tests
More informationNull Hypothesis Significance Testing Signifcance Level, Power, ttests. 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom
Null Hypothesis Significance Testing Signifcance Level, Power, ttests 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom Simple and composite hypotheses Simple hypothesis: the sampling distribution is
More informationHypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...
Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................
More informationHypothesis testing  Steps
Hypothesis testing  Steps Steps to do a twotailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =
More informationIntroduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.
Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.
More informationModule 7: Hypothesis Testing I Statistics (OA3102)
Module 7: Hypothesis Testing I Statistics (OA3102) Professor Ron Fricker Naval Postgraduate School Monterey, California Reading assignment: WM&S chapter 10.110.5 Revision: 212 1 Goals for this Module
More informationMATH 10: Elementary Statistics and Probability Chapter 9: Hypothesis Testing with One Sample
MATH 10: Elementary Statistics and Probability Chapter 9: Hypothesis Testing with One Sample Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of
More informationSampling and Hypothesis Testing
Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus
More information9Tests of Hypotheses. for a Single Sample CHAPTER OUTLINE
9Tests of Hypotheses for a Single Sample CHAPTER OUTLINE 91 HYPOTHESIS TESTING 91.1 Statistical Hypotheses 91.2 Tests of Statistical Hypotheses 91.3 OneSided and TwoSided Hypotheses 91.4 General
More informationLecture 8 Hypothesis Testing
Lecture 8 Hypothesis Testing Fall 2013 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milton Stewart School of Industrial Systems & Engineering Georgia Tech Midterm 1 Score 46 students Highest score: 98 Lowest
More informationChapter 4 Statistical Inference in Quality Control and Improvement. Statistical Quality Control (D. C. Montgomery)
Chapter 4 Statistical Inference in Quality Control and Improvement 許 湘 伶 Statistical Quality Control (D. C. Montgomery) Sampling distribution I a random sample of size n: if it is selected so that the
More informationChapter 7 Notes  Inference for Single Samples. You know already for a large sample, you can invoke the CLT so:
Chapter 7 Notes  Inference for Single Samples You know already for a large sample, you can invoke the CLT so: X N(µ, ). Also for a large sample, you can replace an unknown σ by s. You know how to do a
More informationBasic Elements of a Hypothesis Test. Hypothesis Testing of Proportions and Small Sample Means. Proportions. Proportions
Hypothesis Testing of Proportions and Small Sample Means Dr. Tom Ilvento FREC 408 Basic Elements of a Hypothesis Test H 0 : H a : : : Proportions The Pepsi Challenge asked soda drinkers to compare Diet
More informationChapter 1 Hypothesis Testing
Chapter 1 Hypothesis Testing Principles of Hypothesis Testing tests for one sample case 1 Statistical Hypotheses They are defined as assertion or conjecture about the parameter or parameters of a population,
More informationMAT X Hypothesis Testing  Part I
MAT 2379 3X Hypothesis Testing  Part I Definition : A hypothesis is a conjecture concerning a value of a population parameter (or the shape of the population). The hypothesis will be tested by evaluating
More information15.0 More Hypothesis Testing
15.0 More Hypothesis Testing 1 Answer Questions Type I and Type II Error Power Calculation Bayesian Hypothesis Testing 15.1 Type I and Type II Error In the philosophy of hypothesis testing, the null hypothesis
More informationHypothesis testing allows us to use a sample to decide between two statements made about a Population characteristic.
Hypothesis Testing Hypothesis testing allows us to use a sample to decide between two statements made about a Population characteristic. Population Characteristics are things like The mean of a population
More informationHypothesis testing. Power of a test. Alternative is greater than Null. Probability
Probability February 14, 2013 Debdeep Pati Hypothesis testing Power of a test 1. Assuming standard deviation is known. Calculate power based on onesample z test. A new drug is proposed for people with
More informationBA 275 Review Problems  Week 6 (10/30/0611/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394398, 404408, 410420
BA 275 Review Problems  Week 6 (10/30/0611/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394398, 404408, 410420 1. Which of the following will increase the value of the power in a statistical test
More informationMultiple random variables
Multiple random variables Multiple random variables We essentially always consider multiple random variables at once. The key concepts: Joint, conditional and marginal distributions, and independence of
More informationLAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.
More informationHypothesis Testing or How to Decide to Decide Edpsy 580
Hypothesis Testing or How to Decide to Decide Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at UrbanaChampaign Hypothesis Testing or How to Decide to Decide
More informationHypothesis Testing  One Mean
Hypothesis Testing  One Mean A hypothesis is simply a statement that something is true. Typically, there are two hypotheses in a hypothesis test: the null, and the alternative. Null Hypothesis The hypothesis
More informationSection 12.2, Lesson 3. What Can Go Wrong in Hypothesis Testing: The Two Types of Errors and Their Probabilities
Today: Section 2.2, Lesson 3: What can go wrong with hypothesis testing Section 2.4: Hypothesis tests for difference in two proportions ANNOUNCEMENTS: No discussion today. Check your grades on eee and
More informationIntroduction to Hypothesis Testing
I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters  they must be estimated. However, we do have hypotheses about what the true
More informationChapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing
Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing 83 Testing a Claim About a Proportion 85 Testing a Claim About a Mean: s Not Known 86 Testing
More informationChapter 9: Hypothesis Testing Sections
Chapter 9: Hypothesis Testing Sections Skip: 9.2 Testing Simple Hypotheses Skip: 9.3 Uniformly Most Powerful Tests Skip: 9.4 TwoSided Alternatives 9.5 The t Test 9.6 Comparing the Means of Two Normal
More informationHypothesis Testing. Hypothesis Testing
Hypothesis Testing Daniel A. Menascé Department of Computer Science George Mason University 1 Hypothesis Testing Purpose: make inferences about a population parameter by analyzing differences between observed
More informationHypothesis Testing. Hypothesis Testing CS 700
Hypothesis Testing CS 700 1 Hypothesis Testing! Purpose: make inferences about a population parameter by analyzing differences between observed sample statistics and the results one expects to obtain if
More informationBusiness Statistics, 9e (Groebner/Shannon/Fry) Chapter 9 Introduction to Hypothesis Testing
Business Statistics, 9e (Groebner/Shannon/Fry) Chapter 9 Introduction to Hypothesis Testing 1) Hypothesis testing and confidence interval estimation are essentially two totally different statistical procedures
More informationEstimation of σ 2, the variance of ɛ
Estimation of σ 2, the variance of ɛ The variance of the errors σ 2 indicates how much observations deviate from the fitted surface. If σ 2 is small, parameters β 0, β 1,..., β k will be reliably estimated
More informationCHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING
CHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING MULTIPLE CHOICE 56. In testing the hypotheses H 0 : µ = 50 vs. H 1 : µ 50, the following information is known: n = 64, = 53.5, and σ = 10. The standardized
More informationChapter 9: Hypothesis Testing Sections
Chapter 9: Hypothesis Testing Sections 9.1 Problems of Testing Hypotheses Skip: 9.2 Testing Simple Hypotheses Skip: 9.3 Uniformly Most Powerful Tests Skip: 9.4 TwoSided Alternatives 9.6 Comparing the
More informationConfidence Intervals (Review)
Intro to Hypothesis Tests Solutions STATUB.0103 Statistics for Business Control and Regression Models Confidence Intervals (Review) 1. Each year, construction contractors and equipment distributors from
More informationChapter III. Testing Hypotheses
Chapter III Testing Hypotheses R (Introduction) A statistical hypothesis is an assumption about a population parameter This assumption may or may not be true The best way to determine whether a statistical
More informationHypothesis Testing. Concept of Hypothesis Testing
Quantitative Methods 2013 Hypothesis Testing with One Sample 1 Concept of Hypothesis Testing Testing Hypotheses is another way to deal with the problem of making a statement about an unknown population
More informationIntroduction to Hypothesis Testing OPRE 6301
Introduction to Hypothesis Testing OPRE 6301 Motivation... The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief, or hypothesis, about
More informationChapter 7 Part 2. Hypothesis testing Power
Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship
More informationHypothesis testing for µ:
University of California, Los Angeles Department of Statistics Statistics 13 Elements of a hypothesis test: Hypothesis testing Instructor: Nicolas Christou 1. Null hypothesis, H 0 (always =). 2. Alternative
More information9.1 Basic Principles of Hypothesis Testing
9. Basic Principles of Hypothesis Testing Basic Idea Through an Example: On the very first day of class I gave the example of tossing a coin times, and what you might conclude about the fairness of the
More informationPractice problems for Homework 12  confidence intervals and hypothesis testing. Open the Homework Assignment 12 and solve the problems.
Practice problems for Homework 1  confidence intervals and hypothesis testing. Read sections 10..3 and 10.3 of the text. Solve the practice problems below. Open the Homework Assignment 1 and solve the
More informationNull Hypothesis H 0. The null hypothesis (denoted by H 0
Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property
More informationBasic Statistics Self Assessment Test
Basic Statistics Self Assessment Test Professor Douglas H. Jones PAGE 1 A sodadispensing machine fills 12ounce cans of soda using a normal distribution with a mean of 12.1 ounces and a standard deviation
More informationChapter 9: Hypothesis Testing Sections
Chapter 9: Hypothesis Testing Sections  we are still here Skip: 9.2 Testing Simple Hypotheses Skip: 9.3 Uniformly Most Powerful Tests Skip: 9.4 TwoSided Alternatives 9.5 The t Test 9.6 Comparing the
More information1 Confidence intervals
Math 143 Inference for Means 1 Statistical inference is inferring information about the distribution of a population from information about a sample. We re generally talking about one of two things: 1.
More information6. Statistical Inference: Significance Tests
6. Statistical Inference: Significance Tests Goal: Use statistical methods to check hypotheses such as Women's participation rates in elections in France is higher than in Germany. (an effect) Ethnic divisions
More informationTHE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.
THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM
More information1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96
1 Final Review 2 Review 2.1 CI 1propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years
More informationCHAPTERS 46: Hypothesis Tests Read sections 4.3, 4.5, 5.1.5, Confidence Interval vs. Hypothesis Test (4.3):
CHAPTERS 46: Hypothesis Tests Read sections 4.3, 4.5, 5.1.5, 6.1.3 Confidence Interval vs. Hypothesis Test (4.3): The purpose of a confidence interval is to estimate the value of a parameter. The purpose
More information22. HYPOTHESIS TESTING
22. HYPOTHESIS TESTING Often, we need to make decisions based on incomplete information. Do the data support some belief ( hypothesis ) about the value of a population parameter? Is OJ Simpson guilty?
More informationreductio ad absurdum null hypothesis, alternate hypothesis
Chapter 10 s Using a Single Sample 10.1: Hypotheses & Test Procedures Basics: In statistics, a hypothesis is a statement about a population characteristic. s are based on an reductio ad absurdum form of
More informationStatistics 641  EXAM II  1999 through 2003
Statistics 641  EXAM II  1999 through 2003 December 1, 1999 I. (40 points ) Place the letter of the best answer in the blank to the left of each question. (1) In testing H 0 : µ 5 vs H 1 : µ > 5, the
More informationChapter 21. More About Tests and Intervals. Copyright 2012, 2008, 2005 Pearson Education, Inc.
Chapter 21 More About Tests and Intervals Copyright 2012, 2008, 2005 Pearson Education, Inc. Zero In on the Null Null hypotheses have special requirements. To perform a hypothesis test, the null must be
More informationBasic concepts and introduction to statistical inference
Basic concepts and introduction to statistical inference Anna Helga Jonsdottir Gunnar Stefansson Sigrun Helga Lund University of Iceland (UI) Basic concepts 1 / 19 A review of concepts Basic concepts Confidence
More informationIntroduction. Hypothesis Testing. Hypothesis Testing. Significance Testing
Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters
More informationChapter 9: Hypothesis Testing GBS221, Class April 15, 2013 Notes Compiled by Nicolas C. Rouse, Instructor, Phoenix College
Chapter Objectives 1. Learn how to formulate and test hypotheses about a population mean and a population proportion. 2. Be able to use an Excel worksheet to conduct hypothesis tests about population means
More informationMeasuring the Power of a Test
Textbook Reference: Chapter 9.5 Measuring the Power of a Test An economic problem motivates the statement of a null and alternative hypothesis. For a numeric data set, a decision rule can lead to the rejection
More informationStep 1: Set up hypotheses that ask a question about the population by setting up two opposite statements about the possible value of the parameters.
HYPOTHESIS TEST CLASS NOTES Hypothesis Test: Procedure that allows us to ask a question about an unknown population parameter Uses sample data to draw a conclusion about the unknown population parameter.
More informationHypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam
Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests
More informationHypoTesting. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: Class: Date: HypoTesting Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A Type II error is committed if we make: a. a correct decision when the
More informationMATH 214 (NOTES) Math 214 Al Nosedal. Department of Mathematics Indiana University of Pennsylvania. MATH 214 (NOTES) p. 1/6
MATH 214 (NOTES) Math 214 Al Nosedal Department of Mathematics Indiana University of Pennsylvania MATH 214 (NOTES) p. 1/6 "Pepsi" problem A market research consultant hired by the PepsiCola Co. is interested
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationProbability & Statistics
Probability & Statistics BITS Pilani K K Birla Goa Campus Dr. Jajati Keshari Sahoo Department of Mathematics TEST OF HYPOTHESIS There are many problems in which, rather then estimating the value of a parameter,
More informationSection 7.1. Introduction to Hypothesis Testing. Schrodinger s cat quantum mechanics thought experiment (1935)
Section 7.1 Introduction to Hypothesis Testing Schrodinger s cat quantum mechanics thought experiment (1935) Statistical Hypotheses A statistical hypothesis is a claim about a population. Null hypothesis
More informationC. The null hypothesis is not rejected when the alternative hypothesis is true. A. population parameters.
Sample Multiple Choice Questions for the material since Midterm 2. Sample questions from Midterms and 2 are also representative of questions that may appear on the final exam.. A randomly selected sample
More informationDifference of Means and ANOVA Problems
Difference of Means and Problems Dr. Tom Ilvento FREC 408 Accounting Firm Study An accounting firm specializes in auditing the financial records of large firm It is interested in evaluating its fee structure,particularly
More informationCHAPTER 15: Tests of Significance: The Basics
CHAPTER 15: Tests of Significance: The Basics The Basic Practice of Statistics 6 th Edition Moore / Notz / Fligner Lecture PowerPoint Slides Chapter 15 Concepts 2 The Reasoning of Tests of Significance
More informationHYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1. used confidence intervals to answer questions such as...
HYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
More informationBA 275 Review Problems  Week 5 (10/23/0610/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380394
BA 275 Review Problems  Week 5 (10/23/0610/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380394 1. Does vigorous exercise affect concentration? In general, the time needed for people to complete
More informationTwoSample TTests Assuming Equal Variance (Enter Means)
Chapter 4 TwoSample TTests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one or twosided twosample ttests when the variances of
More informationStatistical Inference: Hypothesis Testing
Statistical Inference: Hypothesis Testing Scott Evans, Ph.D. 1 The Big Picture Populations and Samples Sample / Statistics x, s, s 2 Population Parameters μ, σ, σ 2 Scott Evans, Ph.D. 2 Statistical Inference
More informationConfidence Intervals for One Standard Deviation Using Standard Deviation
Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from
More informationConfidence Intervals for Cp
Chapter 296 Confidence Intervals for Cp Introduction This routine calculates the sample size needed to obtain a specified width of a Cp confidence interval at a stated confidence level. Cp is a process
More informationNull Hypothesis Significance Testing Signifcance Level, Power, ttests Spring 2014 Jeremy Orloff and Jonathan Bloom
Null Hypothesis Significance Testing Signifcance Level, Power, ttests 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom Simple and composite hypotheses Simple hypothesis: the sampling distribution is
More informationIntroduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.
Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative
More informationTwoSample TTests Allowing Unequal Variance (Enter Difference)
Chapter 45 TwoSample TTests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one or twosided twosample ttests when no assumption
More informationHypothesis testing S2
Basic medical statistics for clinical and experimental research Hypothesis testing S2 Katarzyna Jóźwiak k.jozwiak@nki.nl 2nd November 2015 1/43 Introduction Point estimation: use a sample statistic to
More informationExperimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test
Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely
More informationAP Statistics 2002 Scoring Guidelines
AP Statistics 2002 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be sought
More informationHYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1. used confidence intervals to answer questions such as...
HYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
More informationRecall this chart that showed how most of our course would be organized:
Chapter 4 OneWay ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical
More informationChapter Additional: Standard Deviation and Chi Square
Chapter Additional: Standard Deviation and Chi Square Chapter Outline: 6.4 Confidence Intervals for the Standard Deviation 7.5 Hypothesis testing for Standard Deviation Section 6.4 Objectives Interpret
More informationHypothesis testing. c 2014, Jeffrey S. Simonoff 1
Hypothesis testing So far, we ve talked about inference from the point of estimation. We ve tried to answer questions like What is a good estimate for a typical value? or How much variability is there
More informationIntroduction to Hypothesis Testing. Hypothesis Testing. Step 1: State the Hypotheses
Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the
More informationHypothesis testing: Examples. AMS7, Spring 2012
Hypothesis testing: Examples AMS7, Spring 2012 Example 1: Testing a Claim about a Proportion Sect. 7.3, # 2: Survey of Drinking: In a Gallup survey, 1087 randomly selected adults were asked whether they
More informationSTT 430/630/ES 760 Lecture Notes: Chapter 6: Hypothesis Testing 1. February 23, 2009 Chapter 6: Introduction to Hypothesis Testing
STT 430/630/ES 760 Lecture Notes: Chapter 6: Hypothesis Testing 1 February 23, 2009 Chapter 6: Introduction to Hypothesis Testing One of the primary uses of statistics is to use data to infer something
More information1 SAMPLE SIGN TEST. NonParametric Univariate Tests: 1 Sample Sign Test 1. A nonparametric equivalent of the 1 SAMPLE TTEST.
NonParametric Univariate Tests: 1 Sample Sign Test 1 1 SAMPLE SIGN TEST A nonparametric equivalent of the 1 SAMPLE TTEST. ASSUMPTIONS: Data is nonnormally distributed, even after log transforming.
More informationHypothesis Testing Introduction
Hypothesis Testing Introduction Hypothesis: A conjecture about the distribution of some random variables. For example, a claim about the value of a parameter of the statistical model. A hypothesis can
More informationAn Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10 TWOSAMPLE TESTS
The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10 TWOSAMPLE TESTS Practice
More informationIntroduction to Hypothesis Testing
Introduction to Hypothesis Testing A Hypothesis Test for μ Heuristic Hypothesis testing works a lot like our legal system. In the legal system, the accused is innocent until proven guilty. After examining
More information